KP solitons, total positivity, and cluster algebras

Transkript

1 KP solitons, total positivity, and cluster algebras Lauren K. Williams, UC Berkeley (joint work with Yuji Kodama) Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

2 Plan of the talk Understanding regular soliton solutions to the KP equation via total positivity for the Grassmannian and cluster algebras. Background on total positivity on the Grassmannian Background on the KP equation Tight connection between KP solitons and total positivity Application 1: classification of soliton graphs Application 2: connection to cluster algebras Application 3: the inverse problem for soliton graphs Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

3 Total positivity on the Grassmannian The real Grassmannian and its non-negative part The Grassmannian Gr k,n (R) = {V V R n,dimv = k} Represent an element of Gr k,n (R) by a full-rank k n matrix A. Given I ( [n]) k, the Plücker coordinate I (A) is the minor of the k k submatrix of A in column set I. The totally non-negative part of the Grassmannian (Gr k,n ) 0 is the subset of Gr k,n (R) where all Plucker coordinates I (A) are non-negative. Context 1930 s: Beginning of classical theory of totally positive matrices, matrices with all minors positive s: Lusztig developed total positivity in Lie theory. Provided part of the motivation for Fomin-Zelevinsky s cluster algebras : Postnikov defined and studied the totally non-negative part of the real Grassmannian. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

4 Total positivity on the Grassmannian The real Grassmannian and its non-negative part The Grassmannian Gr k,n (R) = {V V R n,dimv = k} Represent an element of Gr k,n (R) by a full-rank k n matrix A. Given I ( [n]) k, the Plücker coordinate I (A) is the minor of the k k submatrix of A in column set I. The totally non-negative part of the Grassmannian (Gr k,n ) 0 is the subset of Gr k,n (R) where all Plucker coordinates I (A) are non-negative. Context 1930 s: Beginning of classical theory of totally positive matrices, matrices with all minors positive s: Lusztig developed total positivity in Lie theory. Provided part of the motivation for Fomin-Zelevinsky s cluster algebras : Postnikov defined and studied the totally non-negative part of the real Grassmannian. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

5 Total positivity on the Grassmannian Postnikov studied (Gr k,n ) 0, found a nice cell decomposition, and showed that cells were in bijection with several interesting combinatorial objects. Theorem (Postnikov) Given a subset M of ( [n]) k, define the positroid cell If S tnn M S tnn M := {A (Gr k,n) 0 I (A) > 0 iff I M}., then it is a cell (homeomorphic to an open ball). We will be interested in only the irreducible positroid cells. (in row-echelon form, its elements don t contain all-zero column or a row which contains all zeros besides the pivot) Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

6 Total positivity on the Grassmannian Theorem (Postnikov) The (irreducible) positroid cells of (Gr k,n ) 0 are in bijection with (and labeled by): Derangements π S n with k excedances (Irreducible) Γ -diagrams L contained in k (n k) rectangle (Irred.) quivalence classes of reduced plabic graphs G of type (k,n) (3,4,5,1,2) = If SM tnn is labeled by the derangement π, we also refer to the cell as Stnn (Similarly for L, G.) Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27 π.

7 Total positivity on the Grassmannian Theorem (Postnikov) The (irreducible) positroid cells of (Gr k,n ) 0 are in bijection with (and labeled by): Derangements π S n with k excedances (Irreducible) Γ -diagrams L contained in k (n k) rectangle (Irred.) quivalence classes of reduced plabic graphs G of type (k,n) (3,4,5,1,2) = If SM tnn is labeled by the derangement π, we also refer to the cell as Stnn (Similarly for L, G.) Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27 π.

8 Total positivity on the Grassmannian Theorem (Postnikov) The (irreducible) positroid cells of (Gr k,n ) 0 are in bijection with (and labeled by): Derangements π S n with k excedances (Irreducible) Γ -diagrams L contained in k (n k) rectangle (Irred.) quivalence classes of reduced plabic graphs G of type (k,n) (3,4,5,1,2) = If SM tnn is labeled by the derangement π, we also refer to the cell as Stnn (Similarly for L, G.) Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27 π.

9 Total positivity on the Grassmannian Theorem (Postnikov) The (irreducible) positroid cells of (Gr k,n ) 0 are in bijection with (and labeled by): Derangements π S n with k excedances (Irreducible) Γ -diagrams L contained in k (n k) rectangle (Irred.) quivalence classes of reduced plabic graphs G of type (k,n) (3,4,5,1,2) = If SM tnn is labeled by the derangement π, we also refer to the cell as Stnn (Similarly for L, G.) Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27 π.

10 The Kadomtsev-Petviashvili equation The KP equation ( 4 u ) x t + 6u u x + 3 u x u y 2 = 0 Proposed by Kadomtsev and Petviashvili in 1970, in order to study the stability of the one-soliton solution of the Korteweg-de Vries (KdV) equation under the influence of weak transverse perturbations. Gives an excellent model to describe shallow water waves Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

11 The Kadomtsev-Petviashvili equation The KP equation ( 4 u ) x t + 6u u x + 3 u x u y 2 = 0 Proposed by Kadomtsev and Petviashvili in 1970, in order to study the stability of the one-soliton solution of the Korteweg-de Vries (KdV) equation under the influence of weak transverse perturbations. Gives an excellent model to describe shallow water waves Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

12 Soliton solutions to the KP equation The real Grassmannian The Grassmannian Gr k,n (R) = {V V R n,dimv = k} Represent an element of Gr k,n (R) by a full-rank k n matrix A. Given I ( [n]) k, I (A) is the minor of the I-submatrix of A. From A Gr k,n (R), can construct τ A, and then a solution u A of the KP equation. (cf Sato, Hirota, Satsuma, Freeman-Nimmo,...) The τ function τ A is given by a Wronskian Fix real parameters κ j such that κ 1 < κ 2 < < κ n. Define j (t 1,...,t n ) := exp(κ j t 1 + κ 2 j t κ n j t n). For J = {j 1,...,j k } [n], define J := j1... jk l<m (κ j m κ jl ). The τ-function is τ A (t 1,t 2,...,t n ) := J (A) J (t 1,t 2,...,t n ). J ( [n] k ) Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

13 Soliton solutions to the KP equation The real Grassmannian The Grassmannian Gr k,n (R) = {V V R n,dimv = k} Represent an element of Gr k,n (R) by a full-rank k n matrix A. Given I ( [n]) k, I (A) is the minor of the I-submatrix of A. From A Gr k,n (R), can construct τ A, and then a solution u A of the KP equation. (cf Sato, Hirota, Satsuma, Freeman-Nimmo,...) The τ function τ A is given by a Wronskian Fix real parameters κ j such that κ 1 < κ 2 < < κ n. Define j (t 1,...,t n ) := exp(κ j t 1 + κ 2 j t κ n j t n). For J = {j 1,...,j k } [n], define J := j1... jk l<m (κ j m κ jl ). The τ-function is τ A (t 1,t 2,...,t n ) := J (A) J (t 1,t 2,...,t n ). J ( [n] k ) Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

14 Soliton solutions to the KP equation The τ function τ A Choose A Gr k,n (R), and fix κ j s such that κ 1 < κ 2 < < κ n. Define j (t 1,...,t n ) := exp(κ j t 1 + κ 2 j t κ n j t n). For J = {j 1,...,j k } [n], define J := j1... jk l<m (κ j m κ jl ). τ A (t 1,t 2,...,t n ) := J ( [n] k ) J(A) J (t 1,t 2,...,t n ). A solution u A (x, y, t) of the KP equation Set x = t 1,y = t 2,t = t 3 (treat other t i s as constants). Then u A (x,y,t) = 2 2 x 2 lnτ A(x,y,t) (1) is a solution to the KP equation. If all I (A) 0, this solution is everywhere regular. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

15 Visualing soliton solutions to the KP equation The contour plot of u A (x, y, t) We analyze u A (x,y,t) by fixing t, and drawing its contour plot C t (u A ) for fixed times t this will approximate the subset of R 2 where u A (x,y,t) takes on its maximum values. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

16 Definition of the contour plot as tropical curve Choose A S tnn M. u A (x,y,t) is defined in terms of τ A (x,y,t) := I M I(A) I (x,y,t). At most points (x,y,t), τ A (x,y,t) will be dominated by one term. Define ˆf A (x,y,t) = max{ J (A) J } J M. Define f A (x,y,t) = max{ln( J (A) J )} J M = max{ln( J (A) (κ jm κ jl )) + i (κ j i x + κ 2 j i y +...)} J M. The contour plot C t (u A ) is the subset of R 2 where f A (x,y,t) is not linear One term I dominates u A in each region of the complement of C t (u A ). Label each region by the dominant exponential. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

17 Definition of the contour plot as tropical curve Choose A S tnn M. u A (x,y,t) is defined in terms of τ A (x,y,t) := I M I(A) I (x,y,t). At most points (x,y,t), τ A (x,y,t) will be dominated by one term. Define ˆf A (x,y,t) = max{ J (A) J } J M. Define f A (x,y,t) = max{ln( J (A) J )} J M = max{ln( J (A) (κ jm κ jl )) + i (κ j i x + κ 2 j i y +...)} J M. The contour plot C t (u A ) is the subset of R 2 where f A (x,y,t) is not linear One term I dominates u A in each region of the complement of C t (u A ). Label each region by the dominant exponential. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

18 Definition of the contour plot as tropical curve Choose A S tnn M. u A (x,y,t) is defined in terms of τ A (x,y,t) := I M I(A) I (x,y,t). At most points (x,y,t), τ A (x,y,t) will be dominated by one term. Define ˆf A (x,y,t) = max{ J (A) J } J M. Define f A (x,y,t) = max{ln( J (A) J )} J M = max{ln( J (A) (κ jm κ jl )) + i (κ j i x + κ 2 j i y +...)} J M. The contour plot C t (u A ) is the subset of R 2 where f A (x,y,t) is not linear One term I dominates u A in each region of the complement of C t (u A ). Label each region by the dominant exponential. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

19 Definition of the contour plot as tropical curve Choose A S tnn M. u A (x,y,t) is defined in terms of τ A (x,y,t) := I M I(A) I (x,y,t). At most points (x,y,t), τ A (x,y,t) will be dominated by one term. Define ˆf A (x,y,t) = max{ J (A) J } J M. Define f A (x,y,t) = max{ln( J (A) J )} J M = max{ln( J (A) (κ jm κ jl )) + i (κ j i x + κ 2 j i y +...)} J M. The contour plot C t (u A ) is the subset of R 2 where f A (x,y,t) is not linear One term I dominates u A in each region of the complement of C t (u A ). Label each region by the dominant exponential. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

20 Definition of the contour plot as tropical curve Choose A S tnn M. u A (x,y,t) is defined in terms of τ A (x,y,t) := I M I(A) I (x,y,t). At most points (x,y,t), τ A (x,y,t) will be dominated by one term. Define ˆf A (x,y,t) = max{ J (A) J } J M. Define f A (x,y,t) = max{ln( J (A) J )} J M = max{ln( J (A) (κ jm κ jl )) + i (κ j i x + κ 2 j i y +...)} J M. The contour plot C t (u A ) is the subset of R 2 where f A (x,y,t) is not linear One term I dominates u A in each region of the complement of C t (u A ). Label each region by the dominant exponential. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

21 Definition of the contour plot as tropical curve Choose A S tnn M. u A (x,y,t) is defined in terms of τ A (x,y,t) := I M I(A) I (x,y,t). At most points (x,y,t), τ A (x,y,t) will be dominated by one term. Define ˆf A (x,y,t) = max{ J (A) J } J M. Define f A (x,y,t) = max{ln( J (A) J )} J M = max{ln( J (A) (κ jm κ jl )) + i (κ j i x + κ 2 j i y +...)} J M. The contour plot C t (u A ) is the subset of R 2 where f A (x,y,t) is not linear One term I dominates u A in each region of the complement of C t (u A ). Label each region by the dominant exponential. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

22 Visualizing soliton solutions to the KP equation Generically, interactions of line-solitons are trivalent or are X-crossings. [6,9] [4,8] [8,9] [2,4] [1,5] 1246 [6,7] [4,7] [4,5] [2,5] [4,8] [1,7] 4589 [7,9] [2,3] [1,5] [1,3] [2,5] [3,7] [6,8] If two adjacent regions are labeled I and J, then J = (I \ {i}) {j}. a The line-soliton between the regions has slope κ i + κ j ; label it [i,j]. a Phase shifts negligable if differences κ i+1 κ i are similar and graph on large scale. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

23 Soliton graphs We associate a soliton graph G t (u A ) to a contour plot C t (u A ) by: forgetting lengths and slopes of edges, and marking a trivalent vertex black or white based on whether it has a unique edge down or up. [6,9] [4,8] [8,9] [2,4] [1,5] 1246 [6,7] [4,7] [4,5] [2,5] [4,8] [1,7] 4589 [7,9] [2,3] [1,5] [1,3] [2,5] [3,7] [6,8] Goal: classify soliton graphs. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

24 Soliton graphs We associate a soliton graph G t (u A ) to a contour plot C t (u A ) by: forgetting lengths and slopes of edges, and marking a trivalent vertex black or white based on whether it has a unique edge down or up. [6,9] [4,8] [8,9] [2,4] [1,5] [6,9] [4,8] [2,4] [1,5] 1246 [1,3] [6,7] [4,8] [4,7] [4,5] [2,5] [1,7] [2,3] [1,5] [2,5] [3,7] 4589 [6,8] [7,9] 1246 [1,3] [2,5] [3,7] 4589 [7,9] [6,8] Goal: classify soliton graphs. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

25 Soliton graphs We associate a soliton graph G t (u A ) to a contour plot C t (u A ) by: forgetting lengths and slopes of edges, and marking a trivalent vertex black or white based on whether it has a unique edge down or up. [6,9] [4,8] [8,9] [2,4] [1,5] [6,9] [4,8] [2,4] [1,5] 1246 [1,3] [6,7] [4,8] [4,7] [4,5] [2,5] [1,7] [2,3] [1,5] [2,5] [3,7] 4589 [6,8] [7,9] 1246 [1,3] [2,5] [3,7] 4589 [7,9] [6,8] Goal: classify soliton graphs. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

26 Total positivity on the Grassmannian and KP solitons [6,9] [4,8] [2,4] [1,5] [7,9] Let A be an element of an (irreducible) positroid cell in (Gr kn ) 0. What can we say about the soliton graph G t (u A )? [1,3] [2,5] [3,7] [6,8] Metatheorem (Kodama-W.) Which cell A lies in determines the asymptotics of G t (u A ) as y ± and t ±. Use the derangement and -diagram labeling the cell. Γ Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

27 Total positivity on the Grassmannian and KP solitons [6,9] [4,8] [2,4] [1,5] [7,9] Let A be an element of an (irreducible) positroid cell in (Gr kn ) 0. What can we say about the soliton graph G t (u A )? [1,3] [2,5] [3,7] [6,8] Metatheorem (Kodama-W.) Which cell A lies in determines the asymptotics of G t (u A ) as y ± and t ±. Use the derangement and -diagram labeling the cell. Γ Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

28 How the positroid cell determines asymptotics at y ± Recall: positroid cells in (Gr kn ) 0 derangements π S n with k exc. Theorem (Biondini-Chakravarty, Chakravarty-Kodama + Kodama-W.) Let A lie in the irreducible positroid cell S tnn π of (Gr kn ) 0. For any t, the soliton graph G t (u A ) has precisely: k line-solitons at y >> 0, labeled by the excedances [i,π(i)] of π, and n k line-solitons at y << 0 labeled by the nonexcedances [i, π(i)]. [6,9] [4,8] [2,4] [1,5] [7,9] [1,3] [2,5] [3,7] [6,8] G t (u A ) where A S tnn π for π = (5,4,1,8,2,9,3, 6, 7). Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

29 How the positroid cell determines asymptotics at t Recall: positroid cells in (Gr k,n ) 0 rectangle Γ -diagrams contained in k (n k) Definition A -diagram is a filling of the boxes of a Young diagram by + s and 0 s such that: there is no 0 with a + above it in the same column, and a + to its left in the same row. Γ Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

30 How the positroid cell determines asymptotics at t Theorem (Kodama-W.) Let L be a -diagram. The following procedure realizes the soliton graph G t (u A ) for A SL tnn and t << 0. Γ Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

31 How the positroid cell determines asymptotics at t Theorem (Kodama-W.) Let L be a -diagram. The following procedure realizes the soliton graph G t (u A ) for A SL tnn and t << 0. Γ Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

32 How the positroid cell determines asymptotics at t Theorem (Kodama-W.) Let L be a -diagram. The following procedure realizes the soliton graph G t (u A ) for A SL tnn and t << 0. Γ Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

33 How the positroid cell determines asymptotics at t Theorem (Kodama-W.) Let L be a -diagram. The following procedure realizes the soliton graph G t (u A ) for A SL tnn and t << 0. Γ Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

34 How the positroid cell determines asymptotics at t Theorem (Kodama-W.) Let L be a -diagram. The following procedure realizes the soliton graph G t (u A ) for A SL tnn and t << 0. Γ Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

35 Plabic graphs and soliton graphs Recall: Positroid cells equivalence classes of reduced plabic graphs. We ve seen that derangements and -diagrams are very useful for understanding soliton graphs. What about plabic graphs? Definition A plabic graph is a planar undirected graph G inside a disk with n boundary vertices 1,..., n in counterclockwise order around the disk s boundary, such that each boundary vertex i is incident to a single edge. Interior vertices are colored black or white. Γ Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

36 Soliton graph generalized plabic graph [6,9] [4,8] [2,4] [1,5] [7,9] [1,3] [2,5] [3,7] [6,8] A generalized plabic graph is a graph in a disk with n boundary vertices 1,...,n labeled in any order around the bdry, s.t. each bdry vertex has degree 1. Two edges may cross. Interior vertices colored black or white. Associate a generalized plabic graph to each soliton graph by: Labeling each bdry vertex incident to the line-soliton [i,π(i)] by π(i). Forgetting the labels of line-solitons and regions. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

37 Soliton graph generalized plabic graph [6,9] [4,8] [2,4] [1,5] [7,9] 7 [1,3] [2,5] [3,7] [6,8] A generalized plabic graph is a graph in a disk with n boundary vertices 1,...,n labeled in any order around the bdry, s.t. each bdry vertex has degree 1. Two edges may cross. Interior vertices colored black or white. Associate a generalized plabic graph to each soliton graph by: Labeling each bdry vertex incident to the line-soliton [i,π(i)] by π(i). Forgetting the labels of line-solitons and regions. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

38 Soliton graph generalized plabic graph [6,9] [4,8] [2,4] [1,5] [7,9] 7 [1,3] [2,5] [3,7] [6,8] A generalized plabic graph is a graph in a disk with n boundary vertices 1,...,n labeled in any order around the bdry, s.t. each bdry vertex has degree 1. Two edges may cross. Interior vertices colored black or white. Associate a generalized plabic graph to each soliton graph by: Labeling each bdry vertex incident to the line-soliton [i,π(i)] by π(i). Forgetting the labels of line-solitons and regions. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

39 Passing from soliton graph generalized plabic graph does not lose any information! Theorem (Kodama-W.) We can reconstruct the labels of line-solitons by following the rules of the road. From the boundary vertex i, turn right at black and left at white. Label each edge along trip with i, and each region to the left of trip by i Consequence: can IDNTIFY the soliton graph with its gen. plabic graph. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

40 Passing from soliton graph generalized plabic graph does not lose any information! Theorem (Kodama-W.) We can reconstruct the labels of line-solitons by following the rules of the road. From the boundary vertex i, turn right at black and left at white. Label each edge along trip with i, and each region to the left of trip by i [6,9] [4,8] [2,4] [1,5] [7,9] [1,3] [2,5] [3,7] [6,8] Consequence: can IDNTIFY the soliton graph with its gen. plabic graph. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

41 Cluster algebras and the Grassmannian Definition (Cluster algebra, cf Fomin and Zelevinsky) A cluster algebra is a kind of commutative ring with a great deal of structure. It has a distinguished family of generators called cluster variables, which are naturally grouped into generating sets called clusters. Can be viewed as kind of discrete dynamical system; close relation with T-systems, Q-systems, etc. Dynamics is governed by quiver mutation. Theorem (J. Scott) The coordinate ring C[Gr k,n ] of the Grassmannian has a natural cluster algebra structure. The Plücker coordinates comprise some of the cluster variables. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

42 Cluster algebras and the Grassmannian Definition (Cluster algebra, cf Fomin and Zelevinsky) A cluster algebra is a kind of commutative ring with a great deal of structure. It has a distinguished family of generators called cluster variables, which are naturally grouped into generating sets called clusters. Can be viewed as kind of discrete dynamical system; close relation with T-systems, Q-systems, etc. Dynamics is governed by quiver mutation. Theorem (J. Scott) The coordinate ring C[Gr k,n ] of the Grassmannian has a natural cluster algebra structure. The Plücker coordinates comprise some of the cluster variables. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

43 Cluster algebras and the Grassmannian Definition (Cluster algebra, cf Fomin and Zelevinsky) A cluster algebra is a kind of commutative ring with a great deal of structure. It has a distinguished family of generators called cluster variables, which are naturally grouped into generating sets called clusters. Can be viewed as kind of discrete dynamical system; close relation with T-systems, Q-systems, etc. Dynamics is governed by quiver mutation. Theorem (J. Scott) The coordinate ring C[Gr k,n ] of the Grassmannian has a natural cluster algebra structure. The Plücker coordinates comprise some of the cluster variables. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

44 Soliton graphs and clusters Definition (The totally positive Grassmannian (Gr k,n ) >0 ) (Gr k,n ) >0 is the set of A Gr k,n s.t. I (A) > 0 I. Theorem (Kodama-W.) Let A (Gr k,n ) >0, and consider the soliton graph G t (u A ). If it is generic (all vertices trivalent), then the set of dominant exponentials labeling G t (u A ) forms a cluster for the cluster algebra associated to the Grassmannian. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

45 Soliton graphs and clusters Definition (The totally positive Grassmannian (Gr k,n ) >0 ) (Gr k,n ) >0 is the set of A Gr k,n s.t. I (A) > 0 I. Theorem (Kodama-W.) Let A (Gr k,n ) >0, and consider the soliton graph G t (u A ). If it is generic (all vertices trivalent), then the set of dominant exponentials labeling G t (u A ) forms a cluster for the cluster algebra associated to the Grassmannian Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

46 Soliton graphs and cluster algebras Corollary Let A (Gr k,n ) >0, and consider the soliton graph G t (u A ). If it is generic then the set of dominant exponentials labeling G t (u A ) represent a set C of algebraically independent Plücker coordinates. Moreover, every Plücker coordinate can be written as a Laurent polynomial in the elements of C. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

47 Application: solving the inverse problem for soliton graphs Inverse problem Given a time t together with the contour plot of a soliton solution of KP, can one reconstruct the point of (Gr k,n ) 0 which gave rise to the solution? Theorem (Kodama-W.) 1. If we know that t << 0 sufficiently small, we can solve the inverse problem, no matter what cell of (Gr k,n ) 0 the element A came from. 2. If the contour plot is generic and came from a point of the TP Grassmannian, we can solve the inverse problem, regardless of time t. Proof of 1: uses our description of soliton graphs at t << 0, and work of Kelli Talaska. Proof of 2: uses result that the set of dominant exponentials labeling such a contour plot forms a cluster in the cluster algebra associated to Gr k,n. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

48 Application: solving the inverse problem for soliton graphs Inverse problem Given a time t together with the contour plot of a soliton solution of KP, can one reconstruct the point of (Gr k,n ) 0 which gave rise to the solution? Theorem (Kodama-W.) 1. If we know that t << 0 sufficiently small, we can solve the inverse problem, no matter what cell of (Gr k,n ) 0 the element A came from. 2. If the contour plot is generic and came from a point of the TP Grassmannian, we can solve the inverse problem, regardless of time t. Proof of 1: uses our description of soliton graphs at t << 0, and work of Kelli Talaska. Proof of 2: uses result that the set of dominant exponentials labeling such a contour plot forms a cluster in the cluster algebra associated to Gr k,n. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

49 Application: solving the inverse problem for soliton graphs Inverse problem Given a time t together with the contour plot of a soliton solution of KP, can one reconstruct the point of (Gr k,n ) 0 which gave rise to the solution? Theorem (Kodama-W.) 1. If we know that t << 0 sufficiently small, we can solve the inverse problem, no matter what cell of (Gr k,n ) 0 the element A came from. 2. If the contour plot is generic and came from a point of the TP Grassmannian, we can solve the inverse problem, regardless of time t. Proof of 1: uses our description of soliton graphs at t << 0, and work of Kelli Talaska. Proof of 2: uses result that the set of dominant exponentials labeling such a contour plot forms a cluster in the cluster algebra associated to Gr k,n. Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

50 Application: classification of soliton graphs for (Gr 2,n ) >0 Up to graph-isomorphism, the soliton graphs for (Gr 2,n ) >0 are in bijection with triangulations of an n-gon Theorem (Kodama-W.) 5 very plabic graph obtained via the above algorithm is a soliton graph G t (u A ) for some A (Gr 2,n ) >0. Conversely, all (generic) soliton graphs for A (Gr 2,n ) >0 can be produced from a triangulation of an n-gon as above. [2,6] 12 [1,3] [1,5] [2,4] [4,6] 45 [3,5] Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

51 Application: classification of soliton graphs for (Gr 2,n ) >0 Up to graph-isomorphism, the soliton graphs for (Gr 2,n ) >0 are in bijection with triangulations of an n-gon Theorem (Kodama-W.) 5 very plabic graph obtained via the above algorithm is a soliton graph G t (u A ) for some A (Gr 2,n ) >0. Conversely, all (generic) soliton graphs for A (Gr 2,n ) >0 can be produced from a triangulation of an n-gon as above. [2,6] 12 [1,3] [1,5] [2,4] [4,6] 45 [3,5] Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

52 Application: classification of soliton graphs for (Gr 2,n ) >0 Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

53 Thanks for listening! (movies?) KP solitons, total positivity, and cluster algebras (K. + W.), PNAS, May 11, KP solitons and total positivity for the Grassmannian (K. + W.), Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27

54 Why look at asymptotics as y ± and not x ±? The equation for a line-soliton separating dominant exponentials I and J is where I = {i,m 2,...,m k } and J = {j,m 2,...,m k } is x + (κ i + κ j )y + (κ 2 i + κ i κ j + κ 2 j )t = constant. So we may have line-solitons parallel to the y-axis, but never to the x-axis. (κ i s are fixed) Lauren K. Williams (UC Berkeley) KP Solitons, total positivity, cluster algebras June / 27